COMPARISION OF REGRESSION WITH NEURAL NETWORK MODEL FOR THE VARIATION OF VANISHING POINT WITH VIEW ANGLE IN DEPTH ESTIMATION WITH VARYING BRIGHTNESS

7/30/2019 COMPARISION OF REGRESSION WITH NEURAL NETWORK MODEL FOR THE VARIATION OF VANISHING POINT WITH VI

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COMPARISION OF REGRESSION WITH NEURAL NETWORK MODEL FOR

THE VARIATION OF VANISHING POINT WITH VIEW ANGLE IN DEPTH

ESTIMATION WITH VARYING BRIGHTNESS

Murali S1,Deepu R

2,Vikram Raju

3

Department of Computer Science and EngineeringMaharaja Institute of Technology, Mysore, Karnataka, [email protected],[email protected],[email protected]

Abstract

A vanishing point in a 2D image is a point where parallel

edges in 3D scene converges due to perspective view.

Vanishing point provides lot of information in most of the

applications like distance of objects in 3D Scene. This

point depends on Camera viewing angle as well as

distance between the parallel edges in a scene. In order

to find relationship between the Vanishing Points withview angle and distance between parallel edges with

varying focal lengths of the camera, various data were

collected by having an experimental set-up by varying the

distance between two parallel sticks in an image and also

by varying the angle of the camera with the parallel

sticks. Images are preprocessed to overcome the

problems of improper lighting in the scene. Different data

mining techniques were used and the results are twofold.

The results of regression model for the relationship

between the angle of the camera and the X and Y

coordinates of the vanishing point has been analyzed

using neural network model, which gave 100% validity on

the model.

Keywords

Vanishing Point, Regression Model, ANOVA, Analysis of

Variance, Image Analysis, Angle Detection, Neural

Networks, Multilayer Perceptron.

Introduction

Vanishing points are useful in applications such as road

detection in unmanned cars, object detection in an image

and training canines etc. In automated vehicles drivingon the roads, it can be used to monitor and control the

vehicle to be kept in the specified lane of the road when itdeviates by tracing the vanishing point. It can also be

used in embedded system applications to assist the

visually impaired to direct them on a pathway. The

method to find the vanishing point works with simple 8-bit gray scale image. The fundamental visual element is

the trend line or directionality of the image data, in the

neighborhood of a pixel. This directionality is measured

as the magnitude of the convolution of a pixelsneighborhood with a Gabor filter. Gabor filters are

directional wavelet-type filters, or masks. They consist of

a plane wave, in any direction in the image plane,

modulated by a Gaussian around a point. The Gaussian

localizes the response and guarantees that the convolutionintegral converges, and the plane wave affords a non-zero

convolution over discontinuities that are perpendicular to

the waves direction. In other directions, the periodicity

of the wave brings the convolution close to zero. The

phase of the wave, relative to a feature at the origin, is

best accounted for by a sine/cosine combination of filters:

cosine responds to even-parity features, sine responds to

odd-parity features, and the other responses are vector

combinations of these two. We are interested only in the

magnitude of the response. The common formulas for the

Gabor filter are:

()

()

where the wave propagates along the x coordinate, which

is rotated in some arbitrary direction relative to the image

coordinates x and y; is the circular frequency of the

wave, = 2 / . Parameter is the half-width of the

Gaussian, and defines roughly the size scale on which

discontinuities are detected. Parameter is the

eccentricity of the Gaussian: if 1, the Gaussian iselongated in the direction of the wave, or perpendicular to

it. Repeated convolutions of the same mask with every

pixels neighborhood can be accelerated by convolving in

the Fourier space, where the convolution of two functions

is expressed as the product of their Fourier transforms.

Intuitively, one can think of the same mask being

translated across the image. In Fourier space, translations

become multiplications by a phase factor, and so it is

possible to transform the image and the mask once,

multiply the transforms, then perform the inversetransform to obtain the value of the convolution at every

pixel of the image at once. In the program used, the image

is convolved with Gabor filters in a (large) number of

evenly spaced directions, from zero to 180 degrees, and

the direction with the strongest response is retained as the

local direction field (trend line) of the image. Naturally,this calculation is by far the most computationally

intensive part of the algorithm [1]. A well-known Fast

Fourier Transform library FFTW is used.

Researchers have proposed various methods to determine

the vanishing point of 2D images. Wolfgang Forstner[5]

presented a method where in the vanishing points were

estimated from line segments and their rotation matrix,

using spherically normalized homogenous coordinates

and allowing vanishing points to be at infinity. The

proposed method had minimal representation foruncertainty of homogenous coordinates of 2D points and
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2D lines and rotations to avoid use of singular covariance

matrices of observed line segments. A more efficient

method was proposed by Danko Antolovic, Alex Leykin,

and Steven D. Johnson, where line segments were filtered

using Gabor filters and having a convolution close to

zero. The output of Gabor filters were processed to Fast

Fourier Transformations. There are many methods to

determine the vanishing point, in this study, a model will

be formed trying to bring out a relationship between theangle of camera and the position of the vanishing points.

Methodology

Numerous photographs were taken by altering the

distance between the sticks and the angle between the

camera and the horizontal. Here the angle () gives theangle between the camera axis and the horizontal. Using

the vanishing point method described by Danko

Antolovic, Alex Leykin, and Steven D. Johnson, the

vanishing point was identified for the various data set

images. The camera angle was varied from 60o to 120o

with increments of 10o, for every alteration in the distance

between the sticks from 25cms to 115cms apart withincrement of 10cms for each trial. The images were taken

for varying intensities for efficient vanishing point

detection. The resultant data was used to build a

relationship between them. A schematic of the

experiment is shown below:

Figure 1: Schematic of the experimental set-up.

PhaseI

General Regression Analysis

A general regression analysis is carried out for the data

set for various angles () and the X and Y coordinates of

the vanishing point determined for the various images.The following table shows the initial results.

Table 1: Regression CoefficientsTerm Coefficient SE Coefficient T P

Constant 231.480 14.8529 15.5848 0.0

X -0.056 0.0122 -4.5650 0.0

Y -0.369 0.0369 -10.0217 0.0

From the above table we get the regression coefficients

for the obtained regression model. The model equation is

defined by,

= 231.480 - 0.056X - 0.369Y

Here, by obtaining the X and Y coordinates of the

vanishing point, the camera angle can be determined.The corresponding P value for the constant, X and Y

coordinates is 0, thus having a strong significance in the

model.

PhaseII

Neural Networks and Multilayer Perceptron

A Multilayer Perceptron (MLP) is a feed forward

artificial neural network model that maps sets of input

data onto a set of appropriate output. MLP utilizes asupervised learning technique called back propagation for

training the network. MLP is a modification of the

standard linear perceptron and can distinguish data that is

not linearly separable.

Input Layer - a vector of predictor variable values (X1 . .. Xp) is presented to the input layer. The input layer

standardizes these values so that the range of each

variable is -1 to 1, using the equation mentioned below.

The input layer distributes the values to each of the

neurons in the hidden layer. In addition to the predictor

variables, there is a constant input of 1.0, called the bias

that is fed to each of the hidden layers, this bias ismultiplied by a weight and added to the sum going into

the neuron.

Hidden Layer - arriving at a neuron in the hidden layer,

the value from each input neuron is multiplied by a

weight (Wji), and the resulting weighted values are added

together producing a combined value Uj. The weighted

sum (Uj) is fed into a transfer function, , which outputs a

value Hj. The outputs from the hidden layer are

distributed to the output layer. In this case since all dataare equally important, all have same weights.

Output Layer-arriving at a neuron in the output layer,the value from each hidden layer neuron is multiplied by

a weight (Wkj), and the resulting weighted values are

added together producing a combined value Vj. Theweighted sum (Vj) is fed into a transfer function, , whichoutputs a value Yk. The Y values are the outputs of the

network.

Missing value handling for this experiment consists of

user and system-missing values are treated as missing.

Statistics are based on cases with valid data for all

variables used by the procedure. Weight handling is not

applicable in this experiment, since all coordinates and

angles and distance between sticks are equally importantand significant to the study.

Multilayer Perceptron: Angle vs. X Coordinate

147 of 210 observations were used to build the models

which were later used to predict the remaining

observations.

Table 2: Case Processing Summary

N Percent

SampleTraining 147 70.0%

Testing 63 30%

Valid 210 100.0%

Excluded 0Total 210

The input layer consists of two covariates, the distancebetween the sticks in the experiment and the X coordinate


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of the vanishing point. It thus consists of two units,

excluding the bias unit in the set-up. The rescaling

method used for the covariates was standardizing. The

dependent and independent variable were standardized

using,

Only one hidden layer is used with three units in the

hidden layer excluding the bias unit. The activation

function used was the hyperbolic tangent function.

The output layer depends only on the X coordinates with

only one unit in it. A standardized rescaling method was

used for the scale dependents. An identity activation

function was used in this case. To incorporate the error, a

sum of squares error function was incorporated.

Synaptic Weight < 0

Synaptic Weight > 0

Figure 2: Multilayer Perceptron model for Angle vs.

X Coordinate

Table 3: Model Summary

Training

Sum of

Squares Error33.206

Relative Error 0.730

Stopping Rule

Used

One consecutive

step(s) with no

decrease in error

(error computationsare based on the

testing sample)

Testing

Sum of

Squares Error8.478


SSE is the error function that the network tries to

minimize during training. Note that the sums of squares

and all following error values are computed for therescaled values of the dependent variables. The relative

error for each scale-dependent variable is the ratio of the

sum-of-squares error for the dependent variable to the

sum-of-squares error for the null model, in which the

mean value of the dependent variable is used as the

predicted value for each case.

Table 4: Parameter Estimates

Predictor

Predicted

Hidden

Layer 1

Output

Layer

H(1:1)X

Coordinates

Input Layer

(Bias) -0.110

Distance 0.038

X -0.821Hidden

Layer 1

(Bias) -0.137

H(1:1) -0.928

Table 5: Independent Variable Importance

Importance Normalized Importance

Distance 0.040 4.2%

X 0.960 100.0%

Figure 3: Variable Importance

Summation of importance is 1. Hence the importance ofeach of the independent variables in predicting thedependent can be understood. From the figure we can

clearly understand the importance of the X coordinates of

the Vanishing point for various camera angles , and hasa 100% importance level, while the distance between the

sticks do not account much for the model. This agrees

with the first regression equation initially derived.

Multilayer Perceptron: Angle vs. Y Coordinate

139 of 210 observations were used to build the model

which was later used to predict 71 of the remaining

observations.


N Percentage

SampleTraining 139 66.1

Testing 71 33.8

Valid 210 100.0

Excluded 0

Total 210

The input layer consists of two covariates, the distance

between the sticks in the experiment and the Y coordinate

of the vanishing point. It thus consists of two units,

excluding the bias unit in the set-up. The rescaling

method used for the covariates was standardizing. Thedependent and independent variable were standardized

using,

Bias

H(1:1)

Bias

Distanc

X

X_Value


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Y_Value

Only one hidden layer was used with three units in the

hidden layer excluding the bias unit. The activation

function used was the hyperbolic tangent function.

The output layer depended only on the Y coordinates with

only one unit in it. A standardized rescaling method wasused for the scale dependents. An identity activation

function was used in this case. To incorporate the error, a

sum of squares error function was incorporated.

Synaptic Weight < 0

Synaptic Weight > 0

Figure 4: Multilayer Perceptron model for

Angle vs.Y Coordinate


Training

Sum of SquaresError

22.622


Stopping Rule

Used

One consecutive

step(s) with no

decrease in error

(error computations

are based on the

testing sample)

Testing

Sum of SquaresError

10.772


Similar to the case of that of the X coordinates, even the

Y coordinates follows the same model.


Predictor

Predicted

Hidden

Layer 1

Output

Layer

H(1:1)Y

Coordinates

Input Layer

(Bias) 0.534

Distance -0.019

Y 0.896

Hidden

Layer 1

(Bias) -0.272

H(1:1) 1.128


Importance Normalized Importance

Distance .019 2.0%

Y .981 100.0%

Summation of importance is 1. Hence the importance of

each of the independent variables in predicting the

dependent can be understood.

Figure 5: Variable Importance

From the figure we can clearly understand the importance

of the Y coordinates of the Vanishing point for variouscamera angles , and has a 100% importance level, while

the distance between the sticks do not account much forthe model. This agrees with the first regression equation

initially derived.

Phase - III

For the same camera angle , the distance between the

sticks, and the brightness intensity was varied at small

intervals to study the effect of the intensity of brightnesson the detection of vanishing points in images. The

results follow.

Multilayer Perceptron: Brightness

X

User and system missing values are treated as missing.

Statistics are based on cases with valid data for allvariables used by the procedure.


N Percent


Testing 68 32.3%

Valid 210 100.0%

Excluded 0

Total 210

Table 11: Network Information

Input

Layer

Covariates

1 Distance

2 X

Number of Unitsa 2

Rescaling Method for

CovariatesStandardized

Hidden

Layer(s)

Number of Hidden

Layers1

Number of Units in

Hidden Layer 1

a

3

Activation Function Hyperbolic tangent

Output

Layer

Dependent

Variables1 X_Value

Bias

Distance

Y

Bias

H(1:1)


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Number of Units 1


Scale DependentsStandardized

Activation Function Identity

Error Function Sum of Squares

a. Excluding the bias unitSynaptic Weight < 0

Synaptic Weight > 0

Figure 6: Hidden layer activation function:

Hyperbolic tangent, Output layer activation function:

Identity


Training

Sum of Squares Error 31.552

Relative Error .725

Stopping Rule Used

1 consecutive

step(s) with no

decrease in errora

Training Time 0:00:00.03

TestingSum of Squares Error 21.762

Relative Error .743

Dependent Variable: X Value

a. Error computations are based on the testing sample.

The error, both sum of squares and relative is high,

implying that higher and lower brightness peaks is not

advisable for the detection of vanishing points.


Predictor Predicted

Hidden Layer 1 Output

LayerH(1:1) H(1:2) H(1:3) X_Value

Input

Layer

(Bias) .086 .280 .007

Distance -.184 .243 -.094

X -.244 .593 -.446

Hidden

Layer 1

(Bias) .083

H(1:1) -.418

H(1:2) .470

H(1:3) -.337


Importance Normalized

Importance

Distance .289 40.7%

X .711 100.0%

Figure 7: Normalized Importance


of the X(Brightness intensity in this case) of the image

under study, and has a 100% importance level, while the

distance between the sticks do account much for the

model, partly due to the relative error.

Multilayer Perceptron: Brightness Y

User and system missing values are treated as missing.

Statistics are based on cases with valid data for allvariables used by the procedure.


N Percent


Testing 59 28%

Valid 210 100.0%

Excluded 0

Total 210

Table 16: Network Information

Input Layer

Covariates1 Distance

2 YNumber of Unitsa 2


CovariatesStandardized

Hidden

Layer(s)

Number of Hidden

Layers1

Number of Units in

Hidden Layer 1a4

Activation FunctionHyperbolic

tangent

Output

Layer

Dependent

Variables1 Y_Value

Number of Units 1


Scale Dependents Standardized

Activation Function Identity

Error FunctionSum of

Squares


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a. Excluding the bias unitSynaptic Weight < 0

Synaptic Weight > 0

Figure 8: Hidden layer activation function:

Hyperbolic tangent, Output layer activation function:

Identity


Training

Sum of Squares Error 27.035Relative Error .581

Stopping Rule Used

1 consecutive

step(s) with no

decrease in errora

Training Time 0:00:00.05

TestingSum of Squares Error 9.178

Relative Error .396

Dependent Variable: Y_Value

a. Error computations are based on the testing sample.

The error, both sum of squares and relative is high,

implying that higher and lower brightness peaks is not

advisable for the detection of vanishing points.


Predictor Predicted

Hidden Layer 1 Output

Layer

H(1:1) H(1:2) H(1:3) H(1:4) Y_Value

Input

Layer

(Bias) 1.311 -.904 -.461 .223

Distance -.127 -.064 .642 .473

Y 1.983 .675 -.767 .284

HiddenLayer 1

(Bias) .470

H(1:1) .894

H(1:2) 1.018

H(1:3) .596

H(1:4) -.342


Importance Normalized

Importance

Distance .119 13.6%

Y .881 100.0%

Figure 9: Normalized Importance


of the Y(Brightness intensity in this case) of the image

under study, and has a 100% importance level, while the

distance between the sticks do account some for the

model, partly due to the relative error.

Conclusion

From the two models, the regression model and the neural

network model, we can easily find a relationship between

the camera angle and the vanishing point. Using this

information and the given vanishing point various

applications can be handled likeroad detection in

unmanned cars, object detection in an image, as a tool toassist the visually challenged, and training canines are a

few to name. Brightness plays an integral part in the

detection of vanishing points, too low and very high

bright photos are not advisable for processing directly.

Pre-processing must be done to improve the edge andhorizon detection, which is yet another emerging field inresearch.

References

[1] Danko Antolovic, Alex Leykin, Steven D.

Johnson, Vanishing Point: a Visual Road-Detection

Program for a DARPA Grand Challenge Vehicle

[2] T. Lee. Image representation using 2D Gabor

wavelets. IEEE Trans. Pattern Analysis and Machine

Intelligence, 18(10):959971, 1996.

[3] M. Sonka, V. Hlavac, R. Boyle, ImageProcessing, Analysis and Machine Vision, (Chapman

& Hall,1993)

[4] M. Nazzal, Ibrahim M. El-Emary and Salam A.

Najim, Multilayer Perceptron Neural Network

(MLPs) For Analyzing the Properties of Jordan Oil

Shale Jamal.

[5] Wolfgang Forstner. Optimal Vanishing Point

Detection and Rotation Estimation of Single Images

from a LegolandScen

[6] D. G. Aguilera, J. Gomez Lahoz and J. FinatCodes A New Method for Vanishing Points

Detection in 3D Reconstruction from A Single View

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COMPARISION OF REGRESSION WITH NEURAL NETWORK MODEL FOR THE VARIATION OF VANISHING POINT WITH VIEW ANGLE IN DEPTH ESTIMATION WITH VARYING BRIGHTNESS